Question

What is Newton's universal law of gravitation? Write it in equation form, and clearly explain what the equation tells us. What do we mean when we say that the law is an inverse square law?

Short answer

Newton's law states that gravitational force is proportional to masses' product and inversely to distance squared.

Step-by-step solution

Understanding Newton's Universal Law of Gravitation

Newton's Universal Law of Gravitation states that every point mass attracts every other point mass in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

Writing the Equation

The mathematical form of Newton's Universal Law of Gravitation is given by the equation: \[ F = G \frac{m_1 m_2}{r^2} \]where:- \( F \) is the gravitational force between the two masses,- \( G \) is the gravitational constant (\(6.674 \times 10^{-11} \text{Nm}^2/\text{kg}^2\)),- \( m_1 \) and \( m_2 \) are the masses of the two objects,- \( r \) is the distance between the centers of the two masses.

Explaining the Equation

The equation tells us that the gravitational force ( F ) is directly proportional to the product of the two masses ( m_1 and m_2 ) involved. It is also inversely proportional to the square of the distance ( r^2 ) between the masses, meaning as the distance increases, the force decreases sharply.

Clarifying the Inverse Square Law

Saying that the law is an inverse square law means that the gravitational force diminishes by the square of the increase in distance. For example, if the distance between the two masses doubles ( r twice), the force becomes one-fourth as strong because ( 1/2^2 = 1/4 ). This inverse square relationship drastically impacts how force behaves with distance changes.

Key concepts

Gravitational Force

Gravitational force is the attractive force acting between any two masses. It is a fundamental interaction in physics, describing how objects with mass pull on each other. This force keeps the planets in orbit around the sun, causes apples to fall to the ground, and holds us firmly on Earth.
To quantify this force, we use the equation: \[ F = G \frac{m_1 m_2}{r^2} \] Here:

• \( F \) represents the gravitational force.
• \( m_1 \) and \( m_2 \) are the masses of the objects in question.
• \( r \) stands for the distance separating the centers of these two masses.
• \( G \) is the gravitational constant, a fixed value that allows us to calculate the intensity of gravitational pull.

Understanding gravitational force helps us appreciate the invisible threads that bind together astronomical bodies and everyday objects alike. Gravitational interactions, minuscule between small masses yet mighty on a celestial scale, illustrate the universal applicability of this force.

Inverse Square Law

The concept of the inverse square law is key to understanding how gravitational force behaves over distance. It posits that a quantity or property varies inversely with the square of the distance from the source of that property or force.
In the context of gravity, this means that the gravitational force becomes weaker as the distance between two objects increases. Specifically, if you double the distance between the two masses, the gravitational force reduces to a quarter of its original strength because the distance squared factor is involved.
The implications of the inverse square law are profound for understanding celestial mechanics and gravitational interactions over large distances. It ensures that even though we may be far from other galactic masses, their influence, albeit small, never entirely vanishes, shaping the fabric of the universe gently and continuously.

Gravitational Constant

The gravitational constant \( G \) is one of the fundamental constants in physics and plays a vital role in the law of gravitation. Its value is approximately \(6.674 \times 10^{-11} \, \text{Nm}^2/\text{kg}^2\). This tiny number indicates how relatively weak the force of gravity is.
Just imagine that with such a small constant, the force can still influence not only earthly bodies but also massive celestial objects over astronomical distances!
The gravitational constant allows us to calculate the gravitational force between any two masses when combined with their masses and the separation distance. It enables us to translate the beautifully succinct equation of gravitational attraction into meaningful predictions and understandings, from the gravitational pull experienced by everyday objects to the forces at play between stars and galaxies.
\( G \) underscores the universality of gravity, integrating it seamlessly with other forces and phenomena, as we observe and measure them in spacetime.